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Central Limit Theorems for Functionals of Persistence Diagrams in Germ-Grain Random Set Models with Applications to Goodness-of-Fit Testing

This paper establishes a central limit theorem for functionals of persistence diagrams in germ-grain random set models using stabilization methods, applying these asymptotic normality results to develop goodness-of-fit tests for detecting spatial interactions in histological breast tissue images.

Original authors: Vesna Gotovac {\DJ}ogaš, Marcela Mandarić

Published 2026-07-13

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CC BY 4.0

Original authors: Vesna Gotovac {\DJ}ogaš, Marcela Mandarić

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to figure out how a crowd of people is behaving in a giant park. Are they huddling together in tight groups (clustering), or are they actively avoiding each other to keep their personal space (repulsion)? In the world of math, this "park" is a random set model, and the "people" are tiny shapes like discs or grains.

For a long time, mathematicians had tools to look at these crowds and guess their behavior, but they lacked a reliable way to say, "I am sure this pattern is different from that one." That's where this paper steps in. The authors, Vesna Gotovac Ðogaš and Marcela Mandarić, have built a new mathematical "super-scope" that proves, with high confidence, that if you look at enough of these random crowds, the patterns they leave behind follow a predictable, bell-shaped curve. This is called a Central Limit Theorem, and it's the golden ticket that turns a hunch into a solid statistical test.

The Magic Map: Persistence Diagrams

To understand the crowd, the authors don't just count heads. They use a technique called Topological Data Analysis (TDA). Imagine taking a photo of the park and turning it into a topographic map. As you slowly raise the water level (a process called "filtration"), islands (connected groups) appear and merge, and lakes (holes) form and fill up.

The authors track every single island and lake, noting exactly when they were "born" (appeared) and when they "died" (merged or filled). They plot these moments on a special map called a Persistence Diagram. On this map:

The x-axis is the birth time.
The y-axis is the death time.
The distance between the two tells you how long the feature lasted.

Features that last a long time are the "important" ones—the big clusters or the massive holes. The authors focus on "M-bounded" features, which is a fancy way of saying they only care about features that aren't infinitely huge or infinitely far apart. This keeps the math manageable.

The Big Discovery: The Bell Curve of Shapes

The paper's main finding is a proof that if you take these persistence diagrams from specific types of random models (called germ-grain models) and calculate certain summaries of them, the results will eventually form a perfect bell curve (a normal distribution) as your observation window gets bigger.

Think of it like rolling dice. If you roll one die, the result is random. But if you roll a million dice and add them up, the total will always follow a predictable bell curve. The authors proved that these complex shape summaries act just like those million dice. This is huge because it means we can now use standard statistical tests to ask: "Is this picture of a tissue sample coming from a 'clustering' model or a 'repulsion' model?"

The Rules of the Game

The authors are very careful about the rules. They proved this works specifically for models where the "people" (the grains) have exponential decay of correlations. In plain English, this means that if two grains are far apart, they don't really care what the other is doing; their influence fades away quickly.

What they ruled out: They didn't just guess this works for any random pattern. They specifically showed it works for models like the Boolean model (randomly scattered discs), Matérn cluster processes (groups of clusters), and determinantal point processes (which naturally repel each other).
What they didn't prove: They didn't claim this works for every possible weird shape or interaction in the universe. They stuck to models where the "memory" of the system dies out quickly over distance.

The Simulation Lab

To see if their new "super-scope" actually works, the authors ran a massive simulation lab. They created digital parks with 7 different types of crowds:

Boolean: Random discs.
Boolean Ellipse: Random ellipses.
Repulsive: Shapes that push each other away.
Cluster: Shapes that hug each other.
Matérn Cluster: Parents with children clusters.
Cell: Shapes arranged in a grid-like cell structure.
DPP: A mathematically "repulsive" pattern.

They then tried to trick the test by feeding it data from one model and asking, "Is this from the Boolean model?"

The Results: The test was a superstar. When the data came from a "Cluster" or "Repulsive" model, the test correctly shouted "NO!" almost 100% of the time.
The Glitch: The test sometimes got confused between the Boolean model and the Boolean Ellipse model. Why? Because the underlying "people" (the centers of the shapes) were arranged exactly the same way in both; only the shape of the "clothes" (circles vs. ellipses) changed. The test, which looks at the gaps between the people, couldn't tell the difference. This wasn't a failure of the math; it was a limitation of the specific question being asked.

Real-World Application: Breast Tissue

Finally, the authors took their method out of the simulation lab and into the real world. They looked at 40 histological images of breast cancer (malignant) and 40 images of mastopathy (benign).

They treated the cells in the images like the "grains" in their models.
They used their new statistical test to see if the cancer tissue looked like the benign tissue.
The Outcome: The test was quite good at distinguishing them. When they assumed the benign tissue was the "normal" model, the test correctly flagged the cancer images as "different" about 92.5% to 100% of the time (depending on the specific measurement). When they flipped it around, it still caught the benign tissue as "different" from the cancer about 85% to 90% of the time.

However, the authors were honest about the limits. They noted that for some of the summary functions they tried, the data didn't quite behave like a perfect bell curve yet. This might be because the real-world images weren't "big enough" (the observation window was small) or because the tissue patterns are just too complex to fit the simple rules perfectly.

The Takeaway

This paper doesn't claim to have solved the mystery of cancer or to have a magic wand that instantly diagnoses disease. Instead, it provides a rigorous, mathematically proven foundation for using shape analysis in statistics. It shows that for a wide class of random patterns, we can now trust our "shape detectors" to tell us if we are looking at a cluster, a repulsion, or something else entirely. It turns the art of looking at patterns into a science with a reliable ruler.

Technical Summary: Central Limit Theorem for Functionals of Persistence Diagrams in Germ-Grain Random Set Models

Problem Statement
Random set models, particularly germ-grain models, are widely used to describe complex geometric structures in fields ranging from ecology to medical histology. While Topological Data Analysis (TDA), specifically through persistence diagrams (PDs), has proven effective as an exploratory tool for distinguishing between different random set models (e.g., detecting clustering or repulsion), a rigorous theoretical foundation for statistical inference based on these diagrams has been lacking. Previous work established methods for testing goodness-of-fit using PDs but did not provide the asymptotic distributional theory required to justify these tests formally. This paper addresses the gap by establishing a Central Limit Theorem (CLT) for functionals of persistence diagrams derived from germ-grain random set models.

Methodology
The authors develop a theoretical framework linking the topological summaries of random sets to probabilistic limit theorems. The core methodology involves the following steps:

Model Definition: The study focuses on germ-grain random set models in $\mathbb{R}^2$ , where a marked point process $\tilde{P} = \{(X_i, M_i)\}$ generates a random set $X = \bigcup (X_i + M_i)$ . The marks $M_i$ are random compact sets (grains), such as discs or ellipses.
Topological Construction: Persistence diagrams are constructed from the signed distance function of the random set realisation. The authors introduce the concept of M-bounded features, restricting the analysis to topological features (connected components and loops) whose birth and death times are determined by local interactions within a bounded diameter $M$ . This localization is crucial for the subsequent probabilistic proofs.
Stabilisation and CLT Derivation: Building on recent advances in stabilisation methods for marked point processes (specifically referencing [26]), the authors define score functions $\xi$ that assign a value to each point in the process based on its local neighborhood. They demonstrate that the integral of a bounded measurable function $f$ over the M-bounded persistence diagram can be expressed as a sum of these score functions.
Conditions for Convergence: The paper proves that if the underlying marked point process exhibits exponential decay of correlations and satisfies specific moment and local dependency conditions (denoted as condition $(\star)$ ), the standardized functional of the persistence diagram converges in distribution to a standard normal random variable as the observation window increases.

Key Contributions

Theoretical Justification: The primary contribution is the proof of a Central Limit Theorem for functionals of M-bounded persistence diagrams arising from germ-grain models. This provides the necessary theoretical underpinning for using PD-based statistics in formal hypothesis testing.
Extension of Stabilisation Theory: The work adapts stabilisation techniques from marked point processes to the specific context of topological summaries, handling the complexity of assigning "birth" and "death" responsibility to specific points in the underlying process.
Goodness-of-Fit Testing Framework: The authors propose two distinct approaches for goodness-of-fit testing based on the derived CLT:
- Rectangular Partitioning: Dividing the persistence diagram into rectangular regions and testing the sum of lifetimes within these regions against a $\chi^2$ distribution.
- Functional Summaries: Utilizing the Accumulated Persistence Function (APF) and the support function of the lift zonoid, which aggregate topological information into scalar or functional summaries suitable for the CLT.

Results
The methodology was validated through a comprehensive simulation study and an application to real-world data:

Simulation Study: The authors simulated various germ-grain models, including Boolean models (Poisson germs), Matérn cluster processes, determinantal point processes (DPP), and Quermass-interaction processes (modeling clustering and repulsion).
- The simulation results confirmed that the proposed test statistics generally follow the asymptotic normality predicted by the CLT.
- In goodness-of-fit tests, the statistics successfully distinguished between models with different interaction patterns (e.g., clustering vs. repulsion) and different underlying point processes.
- The study noted limitations in distinguishing models that share the same underlying point process but differ only in grain shape (e.g., Boolean discs vs. Boolean ellipses) when using dimension 1 features, as the test relies heavily on the spatial arrangement of the germs.
Real-World Application: The framework was applied to histological images of breast tissue, comparing mammary breast cancer (malignant) and mastopathy (benign) tissues.
- Using the rectangular partition method, the tests achieved high rejection rates (up to 100% for dimension 0) when testing one tissue type against the other, demonstrating the ability to distinguish between the two pathological conditions based on their topological signatures.
- The authors noted that functional summaries (APF and lift zonoid support functions) did not exhibit normal behavior for the real data, likely due to the limited size of the observation window or the unknown distribution of the samples, limiting the applicability of those specific summaries in this context.

Significance and Claims
The paper claims that establishing the CLT for persistence diagram functionals bridges the gap between topological data analysis and rigorous statistical inference for random sets. By proving asymptotic normality, the authors enable the use of standard statistical tests (such as $\chi^2$ tests) for model selection and goodness-of-fit, moving beyond purely exploratory TDA.

The authors assert that their TDA-based statistics offer "valuable additional insight" for distinguishing germ-grain models with different interactions and that these new tests are "less time-consuming" than previous global envelope test methods. The successful application to breast tissue histology suggests that these topological summaries can serve as effective biomarkers for differentiating between malignant and benign tissue structures, provided the observation window is sufficiently large to satisfy the asymptotic conditions. The work emphasizes that while the theoretical results rely on exponential decay of correlations, simulation suggests the CLT may hold even when this condition is not strictly met, though this remains an area for further investigation.